Novel Constructions Of Mutually Unbiased Absolutely Maximally Entangled Bases

Quantum entanglement,as an indispensable physical resource,is widely utilized in quantum information processing,and it is one of the hot research issues in the field.Mutually unbiased bases are also useful in quantum secure communication,people have begun to continuously study the mutually unbiased bases based on the entangled states in recent years,and try to apply the classical results in combinatorial designs,graph theory and matrix theory to the study of quantum information theory.In this paper,we mainly study the constructions of mutually unbiased absolutely maximally entangled bases in bipartite and multipartite quantum systems,and draw the following conclusions:(1)In the bipartite system Cd?cd,d-1 maximally entangled bases and a product basis are given when d is a prime power,they are mutually unbiased;mutually unbiased maximally entangled bases and two product bases are given when d is a non-prime power and d?6.(2)If d ?2,6,we put forward a construction of a pair of mutually unbiased absolutely maximally entangled bases in the tripartite system Cd ?Cd? Cd.In particular,if p is a prime number,p-1 mutually unbiased absolutely maximally entangled bases are constructed in the tripartite system Cp?Cp?Cp.(3)By means of direct product structure,for the tripartite quantum system Cd1?Cd2?Cd1d2(2?d1?d2,did2?6),the value of d1d2 is discussed in three cases:(?)d1d2 is a prime power,we get d1d2-1 mutually unbiased absolutely maximally entangled bases;(?)d1d2=2sp1r1p2r2…ptrt(pi?pj?2,piri?3,s?2),we get min{2s-1,p1r1-1,…,ptrt1} mutually unbiased absolutely maximally entangled bases;(?)for other cases,we can construct two mutually unbiased absolutely maximally entangled bases.Moreover,absolutely maximally entangled bases are also mutually unbiased with a product base given by us.(4)We generalize the approach to the quantum system Cd1?Pd2?Cd3?Cd4(d4=d1d2d3,2?d1?d2?d3),and obtain mutually unbiased absolutely maximally entangled bases and a product base.